Geometry:Home Infinity
Infinity is not so much a kind of number, as Cantor posits, but a kind of direction. There are indeed many kinds of infinity, which individually reflect the many kinds of far away. But far away is more a reflection of our ability to get there, rather than what we shall find when we get there.
If we look into the sky, we see faint dots, representing stars. More powerful scopes reveal more about these. But we might note that some dot in our sky, which the most powerful telescope might reveal the largest of planets, is none the less, someone's sun. That is, our perception of these faint stars is more a function of the scope we view them through, and not what is there at the other end.
A continious range is one where any member constructed in the range is also in the set. The proposition exists where there may exist more entities in the range than the set of constructions may reach: that is, there exist points without name.
The notion thas space is made of points is misleading. The name of points
is effected by intersecting n planes (one for each dimension), to mark the
point.
Another notion is that it is possible to pass a line through all points of
a plane. Yet one can show for example, that points like (e, pi) will never
fall on the constructed curve: that is, the cover is discrete.
In terms of space, one has a pair of infinities, or measures.
The gauge infinities derive from the nature of the gauge used to view these. The nature is more a kind of measure of how we can write and compare numbers.
Suppose we have a 10-digit calculator.
By increasing the number of digits on the calculator, and the desire to instance more cases, one can push these numbers further out, but the roles remain the same.
A discrete set is one for which we can for any given range, prove that there exists a nonmember of the set. For example, between any two decimals (B10), lies a number that is not a decimal.
In practice, the number line is peppered with members of the set, and also peppered with non-members. To show that the set is indeed discrete, it suffices to construct for a given range, any member not in the set.
A class-n infinity is a construction which maps an n-dimensional lattice onto a line. Usually this is implemented by way of n incomeasurables. A class-n infinity might often result from the solution of an integral equation to the nth degree, although this is not needed.
The most useful class-infinities arise from the span of chords of a {p} gon. Such is designated as Zp. The class of Zp is half the euler totient of p, eg totient(12) = 4, 4/2 = 2; therefore class{12} = 2. The integer-span for a given odd {p} or even {2p} is Zp. So the integer span for {12} is Z6, since 12 is even, and thus 2p.
Polygons of the same class behave in very similar ways, and often can be mapped onto the same space. The {3}, {4} and {6} are class-one polygons. The pentagons, octagons, decagons, and dodecagons, are all class-two, and all feature a binary isomorphism.
The numbers written in a given base (eg base b), form the set Bb, the set Bb is class-two also. This can be demonstrated both by the notion that one can represent a number and fraction, eg 123.456 by the ordered pair of points, (123, 654). A reversal of the second is required to make the smaller decimals finite, and to distingiush between .01 => 100 and .1 => 10.
More interestingly, like the Bb, one can construct generally, a notation on a class-two system (eg 1, phi), that every number maps uniquely to a single, sequenced expansion. That is, every element of Z5 can be expressed as the sum of unique different, non-adjacent powers of phi, just as one can write every Z2 as a sum of binary powers.
Many discrete sets, such as the fractions or the geometrics, do not have a determined class.
Teel here is from greek telos destination, outcome of journey. The teelic infinities suggest that the different paths formed by the names lead to a smaller set of destinations, and prehaps a finite number of them.
For example, the number we write as
The idea of teelic infinities, is that there might be a suitably large number, eg 71, which describe the complete range of numbers. It is worth noting that one can, in modulo 71, find the chords of the first ten polygons, (except for some of the octagon), and also the usual modulo tricks such as addition, subtraction, division etc. The first six numbers are also square. For a real space of order 6, one might imagine 71 to be a teelic infinity.
Geometry:Home Infinity